kitchentablemath has been hosting a lively discussion on Reform Math's take on children with different learning styles. Reviewing the National Council of Teachers of Mathematics' January, 2008 paper, "Equity in Mathematics Education," SteveH observes the NCTM making such lofty pronouncements as:

The school community acknowledges and embraces all experiences, beliefs, and ways of knowing mathematics.

and:

All students have access to and engage in challenging, rigorous, and meaningful mathematical experiences.

All students have access to and engage in challenging, rigorous, and meaningful mathematical experiences.

In fact, Reform Math proponents display much more interest in celebrating supposed cultural differences in math appreciation than in respecting documented cognitive differences:

Such practices empower all students to build a relationship with mathematics that is positive and grounded in their own cultural roots and history.

The practices in question include a group-centered, hands-on pedagogy that favors a concrete, holistic, social, altogether right-brained learning style. In furthermore requiring all children to explain their answers verbally, these practices downgrade those who solve the typically easy (as compared with traditional math) problems in their heads without thinking them through in words.

Here's an excerpt from an email exchange I had with our school's math consultant, an outspoken Reform Math proponent who holds a Ph.D. in Educational Leadership from one of the top education programs in the country. This exchange took place in January, 2008--the same time as NCTM published its Equity paper.

If I'm remembering correctly, when we were discussing mathematically-gifted but language-impaired kids, you told me that the curriculum allows strategies to be demonstrated in words, numbers, OR, pictures. Is that right? I'm now wondering about problems that use the word "explain", as in "explain [how you got] your answer" rather than "show" ("show how you got your answer"). Is it still the case that the child could answer in numbers (e.g. a series of number sentences) or pictures (e.g. a geometric representation of fractions?)? I'd like to get a better handle on just how well the curriculum accommodates, in particular, the mathematically-gifted but language impaired children that comprise so many of the children on the autistic spectrum, who often literally see the answer, pictographically or numerically, with no accompanying words in their heads.

Math Consultant:

I don't have a simple answer to your question, as it depends somewhat on the problem, concept, grade level & teacher. Most problems do ask the students to show how they got an answer using pictures, numbers, and/or words. There is no set rule as to what is meant by "explain" but the idea would be that someone else should be able to look at the work and know exactly what the student did--often a combination of pictures, numbers and words is necessary to communicate clearly in mathematics. As far as the curriculum is concerned, a central goal is for students to learn to express mathematical thinking through drawing, writing and talking. A teacher would therefore work to develop students' skills in all three areas. Also keep in mind that instruction is also driven by state tests, and on the PSSA students need to explain their thinking in writing on some of the open-ended questions. Therefore teachers need to have students practice this skill throughout the year...

L:

I don't have a simple answer to your question, as it depends somewhat on the problem, concept, grade level & teacher. Most problems do ask the students to show how they got an answer using pictures, numbers, and/or words. There is no set rule as to what is meant by "explain" but the idea would be that someone else should be able to look at the work and know exactly what the student did--often a combination of pictures, numbers and words is necessary to communicate clearly in mathematics. As far as the curriculum is concerned, a central goal is for students to learn to express mathematical thinking through drawing, writing and talking. A teacher would therefore work to develop students' skills in all three areas. Also keep in mind that instruction is also driven by state tests, and on the PSSA students need to explain their thinking in writing on some of the open-ended questions. Therefore teachers need to have students practice this skill throughout the year...

L:

So, to follow up, if the idea is for the teacher to know exactly what the student did, what about a child gets the answer automatically in her head, and simply doesn't know what she herself did in her head (i.e., the answer just "came to her")? In this case, could the student simply write "mental math" as the explanation? A Narberth school allows the mathematically-gifted, non-autistic son of a friend of mine to do this, even on his PSSAs. (Though I don't know whether PSSA graders mark him off for this!)

MC:

Well no, the idea is not just for the teacher to know exactly what the student did, but rather for the student to learn to communicate his or her thinking in a clear way. So "mental math" would not be an adequate explanation. I would certainly think the child would be marked off on the PSSA for that response, given the guidelines that are put out (unless there is some special accommodation in place.) Communication has been a goal of reform mathematics programs since the publication of the standards in 1989.

L:

So if a child doesn't know what his/her thinking was in solving the problem (because it was all subconscious and/or nonverbal thinking), how could this child possibly explain his/her thinking in words? Except to say "I solved it pictographically with the following pictures in my head..." or "it just came to me."

The only alternative I can think of is that such a child would have to imagine how a more verbal person would have solved the problem, and then explain how this hypothetical person would have solved it.

MC:

Well no, the idea is not just for the teacher to know exactly what the student did, but rather for the student to learn to communicate his or her thinking in a clear way. So "mental math" would not be an adequate explanation. I would certainly think the child would be marked off on the PSSA for that response, given the guidelines that are put out (unless there is some special accommodation in place.) Communication has been a goal of reform mathematics programs since the publication of the standards in 1989.

L:

So if a child doesn't know what his/her thinking was in solving the problem (because it was all subconscious and/or nonverbal thinking), how could this child possibly explain his/her thinking in words? Except to say "I solved it pictographically with the following pictures in my head..." or "it just came to me."

The only alternative I can think of is that such a child would have to imagine how a more verbal person would have solved the problem, and then explain how this hypothetical person would have solved it.

MC:

I can certainly appreciate the fact that it would be much more of a challenge for a mathematically gifted but language impaired child, but your question is beyond the realm of my expertise. Perhaps your research will shed some light in this area? I hope I have answered your question about the goals of the curriculum.

L:

Yes, you have.

Unfortunately, these goals, with their narrow notion of what math is about, shortchange the mathematically gifted (ALL those who see the answers nonverbally, including many mathematicians I know). This is, as you know, one big problem I have with the curriculum.

It is too bad that those who have chosen these goals and this curriculum don't seem to know much (or care much?) about how nonverbal children and mathematically inclined curriculum solve math problems.

I can certainly appreciate the fact that it would be much more of a challenge for a mathematically gifted but language impaired child, but your question is beyond the realm of my expertise. Perhaps your research will shed some light in this area? I hope I have answered your question about the goals of the curriculum.

L:

Yes, you have.

Unfortunately, these goals, with their narrow notion of what math is about, shortchange the mathematically gifted (ALL those who see the answers nonverbally, including many mathematicians I know). This is, as you know, one big problem I have with the curriculum.

It is too bad that those who have chosen these goals and this curriculum don't seem to know much (or care much?) about how nonverbal children and mathematically inclined curriculum solve math problems.

Too bad, in particular, for the increasing number of children on the autistic spectrum who are being mainstreamed into regular classroom and whom the education establishment purports to be embracing.

And too bad for the mathematically gifted, whose contributions our mathematically impaired nation must be doing all it can to nurture rather than discourage.

## 6 comments:

This was a fascinating dialogue you have recorded here.

My son has developmental apraxia of speech with receptive and expressive language delays; he is not autistic. I have worked with him to help him symbolize his thoughts and the presumption is, of course, that I myself know what it means to correctly use symbols to express mathematical propositions.

In many ways he picks up on the meaning of symbols quicker than he picks up on natural language. It's inconcievable to me that a curriculum specialist has not taken into account children with language delays since there are so many of them these days.

Yes, indeed. So much for equity.

I'm finding it very interesting how many of us out there blogging on problems with Reform Math have special needs children.

I mostly agree with you, but I would argue that it is even more important to get mathematically gifted children into the habit of showing all work, even if they are able to do it in their heads. As they get older, these are the individuals most likely to study more advanced math, where it becomes very useful to break down operations even when it is possible to do them mentally. I like to compare it to teaching computer programmers to write good comment lines in their code - it's a tiresome annoyance at the lower-levels, but an absolutely crucial step at the higher ones.

I think it makes sense for mathematically gifted students--indeed all students--to show their work for difficult problems. Especially once they reach algebra and beyond, I'm all for it. And I'd be suspicious of anyone who couldn't show their work mathematically.

But Reform Math arithmetic is generally so easy that there's really not much to show, and not much to be learned by showing it. I'm thinking of one of the most recent problems I came across: a 5th grade Investigations problem that asks how you would divide 3/8 of a pizza into two equal parts. Here I think a simple "number sentence" should suffice, and that the child shouldn't have to explain things in words in addition.

My county recently adopted Investigations and our school conducted a math night to inform parents of the program. Although I was unable to attend, my husband went and the one comment he heard over and over from parents was how frustrating it was for their kids to have to draw and write in math. Many of the kids know how to add 4 + 7 and do not need to draw a series of apples to figure it out.

My first grader is taught with TERC math investigations at school, and he does Kumon math at home. So far (thank God) my fourth grader doesn't have to do many investigations. Unfortunately, I see this changing in fifth grade.

Last month, I watched my neighbors's fifth grader walk around from house to house asking those who had cats if he could estimate their length...he was doing a unit in TERC math investigations. How is this going to prepare him for algebra?

Now the Reformists are revising Algebra-- see today's post!

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